Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T17:08:10.030Z Has data issue: false hasContentIssue false
  • English
  • Français

The Rothe method for the wave equation in several space dimensions

Published online by Cambridge University Press:  14 November 2011

Erich Martensen
Erich Martensen
Affiliation:
Mathematisches Institut II, Universität Karlsruhe
Get access Rights & Permissions [Opens in a new window]

Synopsis

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 1-2 , 1979 , pp. 1 - 18
Copyright
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bers, L., John, F. and Schechter, M.. Partial Differential Equations (New York: Interscience Publishers, 1964).Google Scholar
2Garabedian, P. R.. Partial Differential Equations (New York: Wiley, 1964).Google Scholar
3Gerdes, W. and Martensen, E.. Das Rotheverfahren für die räumlich eindimensionale Wellengleichung. Z. Angew. Math. Mech. 58 (1978), T367–T 368.Google Scholar
4Günter, N. M.. Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik (Leipzig: Teubner, 1957).Google Scholar
5Kress, R.. Grundzüge einer Theorie der verallgemeinerten harmonischen Vektorfelder. Meth. Verf. Math. Phys. 2 (1969), 49–83.Google Scholar
6Kress, R.. Die Behandlung zweier Randwertprobleme für die vektorielle Poissongleichung nach einer Integralgleichungsmethode. Arch. Rational Mech. Anal. 39 (1970), 206–226.CrossRefGoogle Scholar
7Kress, R.. Greensche Funktionen und Tensoren für verallgemeinerte harmonische Vektorfelder. Meth. Verf. Math. Phys. 4 (1971), 45–74.Google Scholar
8Ladyshenskaja, O. A., Solonnikov, V. A. and Ural'ceva, N.. Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs 23 (Providence, R. I.: Amer. Math. Soc., 1968).Google Scholar
9Marchuk, G. I.. Methods of numerical mathematics. Applications of Mathematics 2(Berlin: Springer, 1975).Google Scholar
10Martensen, E.. The Convergence of the Horizontal Line Method for Maxwell's Equations. Math. Meth. in the Appl. Sci. 1 (1979), 101–113.CrossRefGoogle Scholar
11Miranda, C.. Partial differential equations of elliptic type. Ergebnisse der Mathematik und ihrer Grenzgebiete 2, 2nd edn (Berlin: Springer, 1970).Google Scholar
12Morrey, C. B.. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften 130 (Berlin: Springer, 1966).Google Scholar
13Rothe, E.. Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math. Ann. 102 (1930), 650–670.CrossRefGoogle Scholar
14Smirnow, W. I.. Lehrgang der hoheren Mathematik 4. Hochschulbücher für Mathematik 5, 7th edn (Berlin: Deutscher Verlag der Wissenschaften, 1975).Google Scholar
15Stummel, F.. Diskrete Konvergenz linearer Operatoren I. Math. Ann. 190 (1970), 45–92.CrossRefGoogle Scholar
16Stummel, F.. Diskrete Konvergenz linearer Operatoren II. Math. Z. 120 (1971), 231–264.CrossRefGoogle Scholar
17Stummel, F.. Diskrete Konvergenz linearer Operatoren III. Linear Operators and Approximation, Butzer, P. L., Kahane, J.-P. and Szökefalvi-Nagy, B. eds. (Basel: Birkhäuser, 1971), 196–216.Google Scholar
18Stummel, F.. Discrete Convergence of Mappings. Topics in Numerical Analysis, Miller, J. ed. (New York: Academic Press, 1973), 285–310.Google Scholar